User blog:Cookiefonster/Analysis of Graham Array Notation - is it well-defined?
I'm sure you all remember Antares I.G. Harrison's Graham Array Notation, a sloppy notation he slapped together and shoved onto the wiki. I've proposed to delete the article due to how little of it is well-defined. Here I'll back up my claims by analyzing the ruleset. Rules 1-3 are alright: :Rule 1 ': a = a :'Rule 2 : a,b = ab :Rule 3 : a^^^.....^^^b with c ^'s using arrow notation. This is a→b→c in chained arrow notation or {a,b,c} in BEAF or BAN. Rule 4 is a bit ambiguous: :Rule 4: a,b,c,d = a{a{.....{a{a{c}b}b}.....}b}b using BEAF's Bracketed Operator It uses the thing of a.....b with c a's or similar, which is generally bad practice. Here he doesn't specify the number of pairs of brackets to put in, so we can only assume it's d pairs of brackets. That does seem to be his intent given how he expresses Graham's number. Rule 5 is alright but superfluous: :Rule 5 : [a,b,c] = a,a,b,c Well-defined, but no need to have a way to truncate an array where the first entry is different. Rules 6-8 are kind of well-defined but very bad practice: Rule 6 : a,b,c,d,e = [a,b,c,d,a,b,c,d,e]http://googology.weebly.com/blog/full-gan *6-1http://googology.wikia.com/wiki/User_blog:Antares.I.G.Harrison/Complete_Graham_Notation (See the "comments" section) : a,b,c,d,e,f,g,h = [a,b,c,d,e,f,a,b,c,d,e,f,g,h] *6-2 : a,b,c,d,e,f,g,h,i = [a,b,c,d,e,f,g,h,a,b,c,d,e,f,g,h,i] *6-3 : a,b,c,d,e,f,g,h,i,j = [a,b,c,d,e,f,g,h,a,b,c,d,e,f,g,h,i,j] *6-4 : a,b,c,d,e,f,g,h,i,j,k = [a,b,c,d,e,f,g,h,i,j,a,b,c,d,e,f,g,h,i,j,k] Rule 7 : a,b,c,d,e,f = [a,b,c,d,a,b,c,d,e,f] Rule 8 : a,b,c,d,e,f,g = [a,b,c,d,e,f,a,b,c,d,e,f,g] At first it just had rules 6, 7, and 8. Here he provides cases for 5, 6, and 7 entry arrays, but not for higher arrays; though it's easy to deduce what to do for 8+ entry arrays, it's better to make a general case, like: :#,z = [#,#,z] which isn't so hard. I suggested that Antares make a rule for arrays with any number of entries, so how did he respond? By adding cases for 8 to 11 entries instead of actually making a general case. He missed the point. Rule 9 doesn't terminate and is superfluous: :Rule 9 : a,(b) = [a,a,(b-1)] :If Harrison is right, Rule 9 is about a^^b+2 This is another weird rule. Why not make a compact expression for an array with b a's and expand upon that? This is just a weaker addition to the notation. It's almost like you're going sideways when you should be going forward. Plus, it doesn't terminate. Let's evaluate 3,(3): 3,(3) = [3,3,(2)] = [3,[3,3,(1)]] = [3,[3,[3,3,(0)]]] = [3,[3,[3,[3,3,(-1)]]]] and so on. It just goes on forever. We can only assume it was intended that either a,(1) or a,(0) evaluates to a. Rule 10 is an expansion on rule 9: :Rule 10 : a,b,(c) = [a,b,a,b,(c-1)] :If Harrison is right, Rule 10 is about a,b,b,c+1 Once again, it's both superfluous and doesn't temrinate. Rule 11 is actally doing a general case right but is still ill-defined: :Rule 11 : #,(a) = [#,#,(a-1)] Rule 12 is weird: :Rule 12 : A 2-dimensional array of a,b (if you can imagine it) like would be a,b^a,b. We will get to the ^ later on. What does "a 2-dimensional array of a,b" mean?! A axb array of a's? A axb array of b's? An array alternating a and b or something? I think we can safely say that's ill-defined. Rule 13 is alright: :Rule 13 : a^b = a,a,a,......a,a (b times) Rule 14 might count as well-defined: :Rule 14 : a^^b = a^a^a^a......^a^a (b times) What would a^b^c even evaluate to? It's definitely not specified. Maybe it would evaluate to [a^b^c]? That seems likely, but it's still ambiguous. Rules 15 and 16 might count as well-defined if you fill in the gaps: :Rule 15 : a{^}b = a^^^......^^^b (b ^'s) :Rule 16 : [a{^}}b = a{^}a{^}a{^}a......{^}a{^}a (b times) Rule 15 just jumps straight to what is clearly extending the a^^b system to up-arrow notation and beyond. Rules 17 and 18 are not well-defined by any stretch of the imagination: :Rule 17 : ab = A c-dimensional array of a,b (BEAF's ab & c) :Rule 18 : ab = A c-dimensional array of a,b with the side lenght d, counting the array as a lenght of 1. This is super ambiguous. Even 2-dimensional arrays don't actually have a clear definition, and how would you extrapolate from there? Would a "3-dimensional array of a,b" be a,b^a,b^a,b?? This is really ambiguous, and kind of meaningless because it doesn't actually represent fully evaluating a cube filled with 3's, rather just a mirage visualization. Rule 19 deserves a flat "what": :Rule 19 : Any array can be ©-ed, be multidimensional, and so on I have nothing to say about this other than that I have no idea what that means—especiialy the ©-ed part. What. Rules 20-22 extrapolate further on something pretty flimsy: :Rule 20 : a,bc = a }.}}}b with c {}'s. :Rule 21 : a,b/c = a,bc :Rule 22 : a,b/c,d = [....[a,b,[a,b,a,b]c]c]....] with d []'s I suppose this could be well-defined? But rule 22 is kind of a case of a salad function. It's applying a weak function to a value (probably) well beyond that function. More coming soon. Category:Blog posts